Calculus of Variations and Geometric Measure Theory

D. Bucur - I. FragalĂ  - A. Giacomini

Multiphase Free Discontinuity Problems: Monotonicity Formula and Regularity Results

created by bucur on 09 Oct 2018


Submitted Paper

Inserted: 9 oct 2018
Last Updated: 9 oct 2018

Year: 2018


The purpose of this paper is to analyze regularity properties of local solutions to free discontinuity problems characterized by the presence of multiple phases. Local solutions are meant according to an ad hoc, nonstandard notion of {\it multiphase local almost-quasi minimizers}. In particular this notion penalizes, among contacts between two different phases, only those which occur at jump points, leaving for free no-jump interfaces which may occur at the zero level of the corresponding state functions. In this setting, our main result states that the phases are open and the jump set (globally considered for all the phases) is essentially closed and Ahlfors regular. This is the same kind of regularity holding for one phase local almost-quasi minimizers of general free discontinuity problems. To achieve the same target in presence of multiple phases demands to set up new refined tools. They are a multiphase monotonicity formula and a multiphase decay lemma, which extend respectively the corresponding one phase results by Bucur-Luckhaus and De Giorgi-Carriero-Leaci. The proof of the former relies on a sharp collective Sobolev extension result for functions with disjoint supports on a sphere, which may be of independent interest.